| L(s) = 1 | + 1.59·2-s + 2.69·3-s + 0.531·4-s + 1.39·5-s + 4.29·6-s + 3.40·7-s − 2.33·8-s + 4.27·9-s + 2.22·10-s + 3.00·11-s + 1.43·12-s + 4.01·13-s + 5.41·14-s + 3.77·15-s − 4.78·16-s − 5.59·17-s + 6.79·18-s − 4.72·19-s + 0.744·20-s + 9.17·21-s + 4.77·22-s − 2.45·23-s − 6.29·24-s − 3.04·25-s + 6.38·26-s + 3.43·27-s + 1.80·28-s + ⋯ |
| L(s) = 1 | + 1.12·2-s + 1.55·3-s + 0.265·4-s + 0.625·5-s + 1.75·6-s + 1.28·7-s − 0.825·8-s + 1.42·9-s + 0.703·10-s + 0.905·11-s + 0.414·12-s + 1.11·13-s + 1.44·14-s + 0.974·15-s − 1.19·16-s − 1.35·17-s + 1.60·18-s − 1.08·19-s + 0.166·20-s + 2.00·21-s + 1.01·22-s − 0.512·23-s − 1.28·24-s − 0.608·25-s + 1.25·26-s + 0.660·27-s + 0.341·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.977145096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.977145096\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 1.59T + 2T^{2} \) |
| 3 | \( 1 - 2.69T + 3T^{2} \) |
| 5 | \( 1 - 1.39T + 5T^{2} \) |
| 7 | \( 1 - 3.40T + 7T^{2} \) |
| 11 | \( 1 - 3.00T + 11T^{2} \) |
| 13 | \( 1 - 4.01T + 13T^{2} \) |
| 17 | \( 1 + 5.59T + 17T^{2} \) |
| 19 | \( 1 + 4.72T + 19T^{2} \) |
| 23 | \( 1 + 2.45T + 23T^{2} \) |
| 29 | \( 1 + 0.534T + 29T^{2} \) |
| 31 | \( 1 + 8.47T + 31T^{2} \) |
| 37 | \( 1 - 4.64T + 37T^{2} \) |
| 41 | \( 1 + 2.25T + 41T^{2} \) |
| 47 | \( 1 + 0.893T + 47T^{2} \) |
| 53 | \( 1 + 9.68T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 2.23T + 61T^{2} \) |
| 67 | \( 1 - 3.27T + 67T^{2} \) |
| 71 | \( 1 - 7.07T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 2.34T + 79T^{2} \) |
| 83 | \( 1 - 3.50T + 83T^{2} \) |
| 89 | \( 1 + 0.915T + 89T^{2} \) |
| 97 | \( 1 - 1.91T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.994326851149099726777342580069, −8.603484392202881798321060903992, −7.88068759780910342851688304966, −6.67177190253654904319446496242, −6.00372335293717998265507161156, −4.90805406372146058836975196506, −4.05271419667487143810929263352, −3.65481897979536887873851404864, −2.27840198703594041946276467219, −1.77802480389169138121023281421,
1.77802480389169138121023281421, 2.27840198703594041946276467219, 3.65481897979536887873851404864, 4.05271419667487143810929263352, 4.90805406372146058836975196506, 6.00372335293717998265507161156, 6.67177190253654904319446496242, 7.88068759780910342851688304966, 8.603484392202881798321060903992, 8.994326851149099726777342580069
